Optimal allocations: Case study / Simulations
2022-10-13
1 Preface
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source("C:/Users/mbofi/Dropbox/CeMSIIS/GitHub/Allocation/case-study/aux_functions.R")2 Case Study
We illustrate the optimal allocations in platform trials by means of a phase II placebo-controlled trial in primary hypercholesterolemia.
In the original study, patients were randomised to the three arms following 1:1:1. In what follows, we used this trial as a motivating study to describe how the patients would have been allocated to the different arms and periods using three allocation strategies -namely, equal allocation (1:…:1), square root of \(k\) (1:…:\(\sqrt(k)\)) and the proposed optimal allocations-, and according to three different trial designs:
- Design with one period only (that is, multi-arm design)
- Design with two periods (arm 2 starts later, but arms 1 and 2 finish at the same time)
- Design with three periods (arm 2 starts later and finishes after arm 1 does)
We also compare the power and type 1 error by means of simulations where we considered the estimated mean in the control arm in the original study. For comparative purposes, in this case study, we suppose total sample size of \(N=80\) and smaller effect sizes. Also, we considered a trial using concurrent controls only.
# means
mean_control = 17.3/3.5
mean_arm1 = 66.2/3.5
mean_arm2 = 72.3/3.52.1 Design 1: multi-arm design
In this case, we consider a design with one period only. The scheme of the trial over time is:
db1_one = sim_designs(r1=1,r2=0,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="one")
db1_sqrt = sim_designs(r1=1,r2=0,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="sqrt")
db1_opt = sim_designs(r1=1,r2=0,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="opt")
plot_trial(db1_one$data$treatment) 
Figure: Design 1: multi-arm design.
# sample sizes
db1_one$ss
#> [,1] [,2] [,3]
#> [1,] 27 0 0
#> [2,] 27 0 0
#> [3,] 27 0 0
db1_sqrt$ss
#> [,1] [,2] [,3]
#> [1,] 23 0 0
#> [2,] 23 0 0
#> [3,] 33 0 0
db1_opt$ss
#> [,1] [,2] [,3]
#> [1,] 23 0 0
#> [2,] 23 0 0
#> [3,] 33 0 0
db1_one_ss <- data.frame(arms=c("A1","A2","C"),db1_one$ss, c(sum(db1_one$ss[1,]),sum(db1_one$ss[2,]),sum(db1_one$ss[3,])))
db1_sqrt_ss <- data.frame(arms=c("A1","A2","C"), db1_sqrt$ss, c(sum(db1_sqrt$ss[1,]),sum(db1_sqrt$ss[2,]),sum(db1_sqrt$ss[3,])))
db1_opt_ss <- data.frame(arms=c("A1","A2","C"), db1_opt$ss, c(sum(db1_opt$ss[1,]),sum(db1_opt$ss[2,]),sum(db1_opt$ss[3,])))The sample sizes per arm and period according to the allocation strategies are the following:
knitr::kable(db1_one_ss, format = "markdown", caption = c("Sample sizes per period and arm (1:1)"), col.names = c("Arms", "Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 27 | 0 | 0 | 27 |
| A2 | 27 | 0 | 0 | 27 |
| C | 27 | 0 | 0 | 27 |
knitr::kable(db1_sqrt_ss, format = "markdown", caption = c("Sample sizes per period and arm (sqrt(k)-rule)"), col.names = c("Arms","Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 23 | 0 | 0 | 23 |
| A2 | 23 | 0 | 0 | 23 |
| C | 33 | 0 | 0 | 33 |
knitr::kable(db1_opt_ss, format = "markdown", caption = c("Sample sizes per period and arm (optimal allocations)"), col.names = c("Arms","Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 23 | 0 | 0 | 23 |
| A2 | 23 | 0 | 0 | 23 |
| C | 33 | 0 | 0 | 33 |
Comparing groups when using 1:1 allocation
res1_one = do.call(rbind.data.frame, models_cc(data = db1_one$data) )
knitr::kable(res1_one, format = "markdown") | p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 16.01093 | 13.41777 | 18.60408 | TRUE | a1 |
| 0 | 15.55270 | 12.85205 | 18.25335 | TRUE | a2 |
Comparing groups when using \(\sqrt(k)\)-allocation (and thus optimal allocations)
res1_opt = do.call(rbind.data.frame, models_cc(data = db1_opt$data) )
knitr::kable(res1_opt, format = "markdown")| p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 13.25836 | 10.46643 | 16.05029 | TRUE | a1 |
| 0 | 16.49544 | 13.96783 | 19.02306 | TRUE | a2 |
2.2 Design 2: two-period design
N = 80
N1 = round(N/4)
N2 = round(N-N1)
c(N1,N2,N-N1-N2)
#> [1] 20 60 0In this case, we consider a design with two periods. The scheme of the trial over time is:
db2_one=sim_designs(r1=N1/N,r2=N2/N,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="one")
db2_sqrt=sim_designs(r1=N1/N,r2=N2/N,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="sqrt")
db2_opt=sim_designs(r1=N1/N,r2=N2/N,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="opt")
plot_trial(db2_one$data$treatment) 
Figure: Design 2: two-period design.
# sample sizes
db2_one$ss
#> [,1] [,2] [,3]
#> [1,] 0 20 0
#> [2,] 10 20 0
#> [3,] 10 20 0
db2_sqrt$ss
#> [,1] [,2] [,3]
#> [1,] 0 18 0
#> [2,] 10 18 0
#> [3,] 10 25 0
db2_opt$ss
#> [,1] [,2] [,3]
#> [1,] 0 24 0
#> [2,] 10 10 0
#> [3,] 10 26 0
db2_one_ss <- data.frame(arms=c("A1","A2","C"),db2_one$ss, c(sum(db2_one$ss[1,]),sum(db2_one$ss[2,]),sum(db2_one$ss[3,])))
db2_sqrt_ss <- data.frame(arms=c("A1","A2","C"), db2_sqrt$ss, c(sum(db2_sqrt$ss[1,]),sum(db2_sqrt$ss[2,]),sum(db2_sqrt$ss[3,])))
db2_opt_ss <- data.frame(arms=c("A1","A2","C"), db2_opt$ss, c(sum(db2_opt$ss[1,]),sum(db2_opt$ss[2,]),sum(db2_opt$ss[3,])))The sample sizes per arm and period according to the allocation strategies are the following:
knitr::kable(db2_one_ss, format = "markdown", caption = c("Sample sizes per period and arm (1:1)"), col.names = c("Arms", "Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 0 | 20 | 0 | 20 |
| A2 | 10 | 20 | 0 | 30 |
| C | 10 | 20 | 0 | 30 |
knitr::kable(db2_sqrt_ss, format = "markdown", caption = c("Sample sizes per period and arm (sqrt(k)-rule)"), col.names = c("Arms","Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 0 | 18 | 0 | 18 |
| A2 | 10 | 18 | 0 | 28 |
| C | 10 | 25 | 0 | 35 |
knitr::kable(db2_opt_ss, format = "markdown", caption = c("Sample sizes per period and arm (optimal allocations)"), col.names = c("Arms","Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 0 | 24 | 0 | 24 |
| A2 | 10 | 10 | 0 | 20 |
| C | 10 | 26 | 0 | 36 |
Comparing groups when using 1:1 allocation
res2_one = do.call(rbind.data.frame, models_cc(data = db2_one$data) )
knitr::kable(res2_one, format = "markdown") | p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 13.08101 | 11.49865 | 14.66336 | TRUE | a1 |
| 0 | 15.87080 | 13.54438 | 18.19723 | TRUE | a2 |
Comparing groups when using \(\sqrt(k)\)-allocation
res2_sqrt = do.call(rbind.data.frame, models_cc(data = db2_sqrt$data) )
knitr::kable(res2_sqrt, format = "markdown")| p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 13.26413 | 11.66374 | 14.86453 | TRUE | a1 |
| 0 | 14.34197 | 11.94262 | 16.74132 | TRUE | a2 |
Comparing groups when using the optimal allocations
res2_opt = do.call(rbind.data.frame, models_cc(data = db2_opt$data) )
knitr::kable(res2_opt, format = "markdown")| p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 13.55084 | 11.84782 | 15.25386 | TRUE | a1 |
| 0 | 16.23535 | 14.29386 | 18.17685 | TRUE | a2 |
2.3 Design 3: three-period design
Suppose now a design with three periods with \(N_1=31\) and consider two situations for \(N_2\), say \(N_2=N-N_1\) and \(N_2= N_1/2\).
2.3.1 Trial with equal allocation rates for periods 1 and 3
Suppose now that the size of the periods are:
N1 = round(N/3)
N2 = round(N-2*N1)
c(N, N1, N2, N-N1-N2)
#> [1] 80 27 26 27Note that in this case the duration of periods 1 and 3 is the same, leading to a symmetrical trial. Below we illustrate the scheme of the trial over time.
db3_one=sim_designs(r1=N1/N,r2=N2/N,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="one")
db3_sqrt=sim_designs(r1=N1/N,r2=N2/N,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="sqrt")
db3_opt=sim_designs(r1=N1/N,r2=N2/N,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="opt")
plot_trial(db3_opt$data$treatment) 
Design 3: three-period design (r1=r3).
# sample sizes
db3_one$ss
#> [,1] [,2] [,3]
#> [1,] 0 9 13
#> [2,] 14 9 0
#> [3,] 14 9 13
db3_sqrt$ss
#> [,1] [,2] [,3]
#> [1,] 0 8 13
#> [2,] 14 8 0
#> [3,] 14 11 13
db3_opt$ss
#> [,1] [,2] [,3]
#> [1,] 0 8 13
#> [2,] 14 8 0
#> [3,] 14 11 13
db3_one_ss <- data.frame(arms=c("A1","A2","C"),db3_one$ss, c(sum(db3_one$ss[1,]),sum(db3_one$ss[2,]),sum(db3_one$ss[3,])))
db3_sqrt_ss <- data.frame(arms=c("A1","A2","C"), db3_sqrt$ss, c(sum(db3_sqrt$ss[1,]),sum(db3_sqrt$ss[2,]),sum(db3_sqrt$ss[3,])))
db3_opt_ss <- data.frame(arms=c("A1","A2","C"), db3_opt$ss, c(sum(db3_opt$ss[1,]),sum(db3_opt$ss[2,]),sum(db3_opt$ss[3,])))The sample sizes per arm and period according to the allocation strategies are the following:
knitr::kable(db3_one_ss, format = "markdown", caption = c("Sample sizes per period and arm (1:1)"), col.names = c("Arms", "Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 0 | 9 | 13 | 22 |
| A2 | 14 | 9 | 0 | 23 |
| C | 14 | 9 | 13 | 36 |
knitr::kable(db3_sqrt_ss, format = "markdown", caption = c("Sample sizes per period and arm (sqrt(k)-rule)"), col.names = c("Arms","Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 0 | 8 | 13 | 21 |
| A2 | 14 | 8 | 0 | 22 |
| C | 14 | 11 | 13 | 38 |
knitr::kable(db3_opt_ss, format = "markdown", caption = c("Sample sizes per period and arm (optimal allocations)"), col.names = c("Arms","Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 0 | 8 | 13 | 21 |
| A2 | 14 | 8 | 0 | 22 |
| C | 14 | 11 | 13 | 38 |
Comparing groups when using 1:1 allocation
res3_one = do.call(rbind.data.frame, models_cc(data = db3_one$data) )
knitr::kable(res3_one, format = "markdown") | p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 13.77077 | 12.68510 | 14.85644 | TRUE | a1 |
| 0 | 14.55572 | 13.49474 | 15.61669 | TRUE | a2 |
Comparing groups when using \(\sqrt(k)\)-allocation
res3_sqrt = do.call(rbind.data.frame, models_cc(data = db3_sqrt$data) )
knitr::kable(res3_sqrt, format = "markdown")| p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 13.59788 | 12.53378 | 14.66199 | TRUE | a1 |
| 0 | 15.81046 | 14.62007 | 17.00085 | TRUE | a2 |
Comparing groups when using the optimal allocations
res3_opt = do.call(rbind.data.frame, models_cc(data = db3_opt$data) )
knitr::kable(res3_opt, format = "markdown")| p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 13.03789 | 11.80427 | 14.27152 | TRUE | a1 |
| 0 | 15.64012 | 14.52241 | 16.75782 | TRUE | a2 |
2.3.2 Trial with unequal allocation rates for for periods 1 and 3
Suppose now that the size of the periods are:
# N = 80
N1 = round(N/3)
N2 = round(2*(N-N1)/3)
c(N1,N2,N-N1-N2)
#> [1] 27 35 18db3_one=sim_designs(r1=N1/N,r2=N2/N,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="one")
db3_sqrt=sim_designs(r1=N1/N,r2=N2/N,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="sqrt")
db3_opt=sim_designs(r1=N1/N,r2=N2/N,mu0=mean_control,mu1=mean_arm1,mu2=mean_arm2,N=N,alloc="opt")
plot_trial(db3_opt$data$treatment) 
Design 3: three-period design (r1<r3).
# sample sizes
db3_one$ss
#> [,1] [,2] [,3]
#> [1,] 0 12 9
#> [2,] 14 12 0
#> [3,] 14 12 9
db3_sqrt$ss
#> [,1] [,2] [,3]
#> [1,] 0 10 9
#> [2,] 14 10 0
#> [3,] 14 14 9
db3_opt$ss
#> [,1] [,2] [,3]
#> [1,] 0 13 9
#> [2,] 14 7 0
#> [3,] 14 15 9
db3_one_ss <- data.frame(arms=c("A1","A2","C"),db3_one$ss, c(sum(db3_one$ss[1,]),sum(db3_one$ss[2,]),sum(db3_one$ss[3,])))
db3_sqrt_ss <- data.frame(arms=c("A1","A2","C"), db3_sqrt$ss, c(sum(db3_sqrt$ss[1,]),sum(db3_sqrt$ss[2,]),sum(db3_sqrt$ss[3,])))
db3_opt_ss <- data.frame(arms=c("A1","A2","C"), db3_opt$ss, c(sum(db3_opt$ss[1,]),sum(db3_opt$ss[2,]),sum(db3_opt$ss[3,])))The sample sizes per arm and period according to the allocation strategies are the following:
knitr::kable(db3_one_ss, format = "markdown", caption = c("Sample sizes per period and arm (1:1)"), col.names = c("Arms", "Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 0 | 12 | 9 | 21 |
| A2 | 14 | 12 | 0 | 26 |
| C | 14 | 12 | 9 | 35 |
knitr::kable(db3_sqrt_ss, format = "markdown", caption = c("Sample sizes per period and arm (sqrt(k)-rule)"), col.names = c("Arms","Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 0 | 10 | 9 | 19 |
| A2 | 14 | 10 | 0 | 24 |
| C | 14 | 14 | 9 | 37 |
knitr::kable(db3_opt_ss, format = "markdown", caption = c("Sample sizes per period and arm (optimal allocations)"), col.names = c("Arms","Period 1","Period 2","Period 3", "Total per arm"))| Arms | Period 1 | Period 2 | Period 3 | Total per arm |
|---|---|---|---|---|
| A1 | 0 | 13 | 9 | 22 |
| A2 | 14 | 7 | 0 | 21 |
| C | 14 | 15 | 9 | 38 |
Comparing groups when using 1:1 allocation
res3_one = do.call(rbind.data.frame, models_cc(data = db3_one$data) )
knitr::kable(res3_one, format = "markdown") | p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 15.38102 | 14.28894 | 16.47311 | TRUE | a1 |
| 0 | 16.88597 | 15.52827 | 18.24367 | TRUE | a2 |
Comparing groups when using \(\sqrt(k)\)-allocation
res3_sqrt = do.call(rbind.data.frame, models_cc(data = db3_sqrt$data) )
knitr::kable(res3_sqrt, format = "markdown")| p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 13.82508 | 12.70810 | 14.94206 | TRUE | a1 |
| 0 | 15.18625 | 13.96799 | 16.40451 | TRUE | a2 |
Comparing groups when using the optimal allocations
res3_opt = do.call(rbind.data.frame, models_cc(data = db3_opt$data) )
knitr::kable(res3_opt, format = "markdown")| p_val | treat_effect | lower_ci | upper_ci | reject_h0 | arm |
|---|---|---|---|---|---|
| 0 | 13.42321 | 12.20348 | 14.64294 | TRUE | a1 |
| 0 | 16.78099 | 15.58322 | 17.97876 | TRUE | a2 |
3 Simulations
load("C:/Users/mbofi/Dropbox/CeMSIIS/GitHub/Allocation/case-study/results/simstudy_results.RData")
df_res$design = ifelse(as.numeric(df_res$r1)+as.numeric(df_res$r2)==1,"2-period", "3-period")To compare power and type 1 error of the different designs, we undertake a simulation study to evaluate the performance when using 1:1 allocations. For comparative purposes, we also consider a total sample size for the trial equal to XX
res_report_H1 <- df_res %>% filter(H0=="FALSE") %>% select(minrt,rt_a1,rt_a2,r1,r2,alloc,design)
knitr::kable(res_report_H1, format = "markdown", caption = c("Power comparisons"), col.names=c("Min Power", "Power A1", "Power A2", "r1", "r2", "Allocation", "Design"))| Min Power | Power A1 | Power A2 | r1 | r2 | Allocation | Design |
|---|---|---|---|---|---|---|
| 0.90015 | 0.93188 | 0.90015 | 0.3375 | 0.4375 | one | 3-period |
| 0.91773 | 0.91773 | 0.92704 | 0.3375 | 0.4375 | opt | 3-period |
| 0.91739 | 0.94009 | 0.91739 | 0.3375 | 0.4375 | sqrt | 3-period |
| 0.40323 | 0.71062 | 0.40323 | 0.25 | 0.75 | one | 2-period |
| 0.48907 | 0.6354 | 0.48907 | 0.25 | 0.75 | opt | 2-period |
| 0.40952 | 0.70559 | 0.40952 | 0.25 | 0.75 | sqrt | 2-period |
res_report_H0 <- df_res %>% filter(H0=="TRUE") %>% select(minrt,rt_a1,rt_a2,r1,r2,alloc,design)
knitr::kable(res_report_H0, format = "markdown", caption = c("Type 1 error rate"), col.names=c("Min T1E", "T1E A1", "T1E A2", "r1", "r2", "Allocation", "Design"))| Min T1E | T1E A1 | T1E A2 | r1 | r2 | Allocation | Design |
|---|---|---|---|---|---|---|
| 0.0252 | 0.0252 | 0.02542 | 0.3375 | 0.4375 | one | 3-period |
| 0.0245 | 0.02456 | 0.0245 | 0.3375 | 0.4375 | opt | 3-period |
| 0.02396 | 0.02483 | 0.02396 | 0.3375 | 0.4375 | sqrt | 3-period |
| 0.02464 | 0.02464 | 0.02515 | 0.25 | 0.75 | one | 2-period |
| 0.02139 | 0.02139 | 0.02433 | 0.25 | 0.75 | opt | 2-period |
| 0.02425 | 0.02425 | 0.02471 | 0.25 | 0.75 | sqrt | 2-period |
Center for Medical Statistics, Informatics and Intelligent Systems, Medical University of Vienna.
[Klassifizierung: vertraulich]
marta.bofillroig@meduniwien.ac.at
and Martin Posch
martin.posch@meduniwien.ac.at